Can someone tell me the joke/story behind this?

In Fate/Grand Order classes have weaknesses and resistances to other classes, and one of those is Caster > Assassin.

Nursery Rhyme is a Caster, and Soul Eaters (the dog monster) are Assassins, so she is good at killing them. The art was also made during a recent event where females got massive damage boosts so she was a good option for the node with Soul Eaters. The 'critical' is because she is sort of a critical damage orientated servant.

The fifth picture is a parody of Fate/Prototype: Fragment of Silver And Blue's Archer's NP. The NP is described as a lone comet.

The sixth picture is a play on this year April Fool joke for Fate/Grand Order, which replaced every card's art with Riyo's (the artist of Learning FGO manga). The art for Nursery Rhyme has her reading an R-18 book, which became a running joke for the fan.

Assassin is nicknamed the Savior of France because he is that badass and a dragons are just bigger swallows.

The story behind it was that Fate/ Grand Order had dragons in France...and they were hard to kill. Since Sasaki was a one-star (very common, but easy to upgrade), people started to use him a lot to kill the dragons since assassin > rider (the dragon). Thus, he is forever the Savior of France :D.